Texas Instruments UCD3138 - Page 42

Q 0.5£
z1 z2
z1 z2
Q
w ´ w
=
w + w
r z1 z2
w = w ´ w
2
r
z1, z2
1 1 4 Q
2 Q
w
æ ö
w = ± - ´
ç ÷
´
è ø
2
2
r
r
cz 0
p1
s s
1
Q
G (s) K
s
s 1
æ ö
ç ÷
+ +
ç ÷
´ w
w
è ø
=
æ ö
+
ç ÷
ç ÷
w
è ø
z1 z2
cz 0
p1
s s
1 1
G (s) K
s
s 1
æ öæ ö
+ +
ç ÷ç ÷
w w
è øè ø
=
æ ö
+
ç ÷
ç ÷
w
è ø
1 1
c P I D
1 1
1 z 1 z
G (z) K K K
1 z 1 z
- -
- -
+ -
= + +
- - a ´
Digital PFC Description www.ti.com
12.9.1 Loop Compensation from Poles and Zeros in s-Domain
PID control in the UCD3138 CLA for current control loop in single-phase PFC is formed in the following
equation in z-domain:
(3)
If Equation 3 is converted to the s-domain equivalent using the bilinear transform, the result has two
forms. One is with two real zeros:
(4)
The two zeros can also be presented with complex conjugates and in such case,
(5)
Two complex conjugate zeros are expressed as:
(6)
(7)
(8)
The complex conjugate zeros become real zeros when:
(9)
42
Digitally Controlled Single-Phase PFC Pre-Regulator SLUU885B–March 2012–Revised July 2012
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